Calculus of Variations and Geometric Measure Theory

V. De Cicco - G. Scilla

Lower semicontinuity in $GSBD$ for nonautonomous surface integrals

created by scilla on 08 Apr 2022
modified on 25 Jan 2023


Published Paper

Inserted: 8 apr 2022
Last Updated: 25 jan 2023

Journal: ESAIM: COCV
Volume: 29
Number: 13
Year: 2023
Doi: 10.1051/cocv/2023001

ArXiv: 2204.04989 PDF


We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$ : the notion of nonautonomous symmetric joint convexity, which extends the analogous definition devised for autonomous integrands in where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in $SBV$ obtained in, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.

Keywords: Lower Semicontinuity, fracture mechanics, Chain Rule, capacity, $GSBD$ functions